On exact sequences of Hodge theoretic fundamental groups

TL;DR

The paper defines the Hodge theoretic fundamental group $\pi_1(\mathrm{VMHS}(X),F_x)$ as the Tannakian fundamental group of variations of mixed Hodge structure with integral structures and investigates the relative exact sequence $1\to \pi_1(\mathrm{LS}^{\text{Hdg}}(X_b))\to \pi_1(\mathrm{VMHS}(X))\to \pi_1(\mathrm{VMHS}(B))\to 1$ for smooth projective families $f:X\to B$. It proves exactness in the relative setting, studies when the left map fails to be injective, and introduces the non-abelian Hodge locus NAHL and the non-abelian Hodge exceptional locus NG$(X/B)$, conjecturing algebraicity under suitable hypotheses. Through a case study of Pic$^1_{\mathcal{C}_g/\mathcal{M}_g}$, it constructs explicit faithful monodromy local systems and demonstrates exactness in that setting, while linking Hodge theoretic splittings to obstructions from the étale fundamental group. The work further relates the Hodge theoretic section question to étale sections, indicating how étale obstructions can obstruct group-theoretic sections and outlining a framework to compare Hodge-theoretic and étale data. An appendix carries out a key monodromy computation using Johnson homomorphisms and results of Hain–Matsumoto.

Abstract

The goal of this paper is to first define a Hodge theoretic fundamental group for smooth connected complex algebraic varieties and then prove and study a right exact sequence of Hodge theoretic fundamental groups associated to a smooth projective family of algebraic varieties $f\colon X\to B$. In particular, we study when this right exact sequence is exact, relate this question to some prior results in non-abelian Hodge theory, and give an obstruction to splitting in terms of étale fundamental groups. The main examples we consider in this note is the universal curve $f\colon \mathcal{C}_g\to \mathcal{M}_g$ and the moduli space of degree $1$ line bundles on the universal curves $p \colon \text{Pic}^1_{\mathcal{C}_g/\mathcal{M}_g}\to \mathcal{M}_g$.

Theorems & Definitions (61)

• Definition 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.8
• Theorem 1.9
• Theorem 2.1
• Proposition 2.2